Suppose that $M$ is a smooth manifold with a boundary. Let $\mathrm{int}(M):=M\setminus\partial M$ be its interior. Is there a relation between the (relative or absolute) homology of $M$ and the homologies of $\mathrm{int}(M)$ and $\partial M$?
The naive guess $H_k(M) \overset{?}{\cong} H_k(\mathrm{int}(M))\oplus H_k(\partial M)$ doesn't work. E.g. as mentioned by @coudy in the comments, if $M$ is a disk this is obviously false.
On
There is a relation between the relative cohomology of $M$ with respect to its boundary and the homology of its boundary $\partial X$, called Lefschetz duality. Given a compact orientable manifold $X$ of dimension $n$, for all $q$, there is an isomorphism
$$H^q(X,\partial X) \simeq H_{n-q}(X).$$
This can be used, for example, to obtain a relation between the homology of $X$, its boundary $\partial X$ and its double $2X$ obtained by gluing $X$ to itself along its boundary. This implies the following relation between the Euler characteristics, $$ \chi(2X) = 2\chi(X)-\chi(\partial X). $$
The inclusion $\mathrm{int}(M)\rightarrow M$ is a homotopy equivalence. This follows from the existence of a collar neighborhood of $\partial M$ in $M$. Thus, $H_i(\mathrm{int}(M))\rightarrow H_i(M)$ and $H^i(M)\rightarrow H^i(\mathrm{int}(M))$ are isomorphisms for all $i\ge0$. The relation between $H_i(\partial M)$ and $H_i(M)$ is not quite so clear, I think. There is, of course, a long exact pair sequence of the form $\dotsc\rightarrow H_i(\partial M)\rightarrow H_i(M)\rightarrow H_i(M,\partial M)\rightarrow\dotsc$. Furthermore, Lefschetz duality allows you to identify $H_i(M,\partial M)\cong H^{n-i}_c(M)$. Thus, the sequence has one term depending on the homotopy type of $\partial M$, one term depending on the homotopy type of $M$ and one term depending on the proper homotopy type of $M$. If $M$ is compact, then of course $H_c^{n-i}(M)=H^{n-i}(M)$, so two terms each in the sequence only depend on the homotopy type of $M$ and the third term each on the homotopy type of $\partial M$.